# Solving Quadratic Equations by Factorisation

## Solving Quadratic Equations by Factorisation Revision

**Solving Quadratic Equations by Factorisation**

**Quadratic equations** are **equations** that take the form, or can be rearranged to the form,

ax^{2}+bx+c=0

These can be solved by **factorising** the ax^{2}+bx+c part of the equation into **two brackets**, then setting each **bracket** to 0 to derive **two solutions**, called **roots**.

**Note:** Not all **quadratics** can be **factorised**, so not all **quadratics** can be solved with this method.

Make sure you are happy with the following topics before continuing:

**How to Solve a Quadratic Equation by Factorisation**

The general steps to solve a **quadratic** by **factorisation** are as follows:

**Step 1:** If necessary, rearrange the **quadratic** into the form ax^{2}+bx+c=0

**Step 2:** Factorise the **quadratic** into two **brackets** (px+q)(rx+s)=0

**Step 3:** Get two solutions by solving the two linear equations px+q=0 and rx+s=0

**Example:** Solve by **factorisation** x^{2}+4x+3=0

**Step 1:** This **quadratic** is already in the form we want, so we don’t need to rearrange.

**Step 2:** Factorise the **quadratic**: x^{2}+4x+3=(x+3)(x+1)=0

**Step 3:** Set **both brackets** to 0 and solve:

x+3=0\rightarrow x=-3 and x+1=0\rightarrow x=-1

So our **two solutions** are x=-3 and x=-1

**Tip:** For **quadratics** with a=1, we can read the **solutions** off from the **factorisation**, as the **roots** of (x+p)(x+q)=0 are -p and -q. For **quadratics** with a\neq1 it is more complicated.

**Example:** Solve by **factorisation** 2x^{2}-11x=-5

**Step 1:** Rearrange by adding 5 to both sides: 2x^{2}-11x+5=0

**Step 2:** Factorise the **quadratic**: (2x-1)(x-5)=0

**Step 3:** Set **both brackets** to 0 and solve:

2x-1=0\rightarrow x=\dfrac{1}{2} and x-5=0\rightarrow x=5

So our **two solutions** are x=\dfrac{1}{2} and x=5

**Note:** The reason this method to solve **quadratics** works is that, once **factorised**, the quadratic is **two brackets** that **multiply** together to give 0. This can only happen when one of the **brackets** is 0, so we can get **solutions** by assuming that is the case.

**Factorisation and Quadratic Graphs**

We can use **factorisation** to help us sketch a **quadratic** **graph**. This is because the **roots** of the **quadratic** **equation** is where the graph crosses the x-axis, so we can **factorise** to find the **roots** and use these to help determine the correct shape of the graph.

**Example:** Sketch the graph y=x^{2}-2x-3

**Step 1:** For a **graph** we want it to be in the form y=ax^{2}+bx+c, which we already have. We then set y=0.

**Step 2:** **Factorise** the **quadratic**: (x-3)(x+1)=0

**Step 3:** Set **both brackets** to 0 and solve:

x-3=0\rightarrow x=3 and x+1=0\rightarrow x=-1

Alternatively we can use the tip from earlier to read off the **roots** from the **brackets** as x=3 and x=-1

**Step 4:** These **roots** are the x-intercepts of the graph – the places where the **graph** crosses the x-axis. So we know the **graph** crosses the points (3,0) and (-1,0).

**Step 5:** Find the y-intercept. For a **graph** of the form y=ax^{2}+bx+c, this is just the number on the end, c. In this example, it is -3, so the graph also goes through (0,-3).

**Step 6:** Use the points we have found to sketch the **graph**. The **graph** on the right was drawn by a computer, a sketch will not need to look as accurate as this. Instead, a sketch only has to be the right shape and cross the axes at the points we found.

**Example 1: Solving a Quadratic Equation by Factorisation (a=1)**

Solve by **factorisation** x^{2}=7x-12

**[2 marks]**

**Step 1:** Rearrange by subtracting 7x-12 from both sides:

x^{2}-7x+12=0

**Step 2:** **Factorise** the **quadratic**:

x^{2}-7x+12=(x-4)(x-3)=0

**Step 3:** Find the **solutions** by setting **both brackets** equal to 0:

x-4=0\rightarrow x=4 and x-3=0\rightarrow x=3

Alternatively, since a=1, the solutions x=4 and x=3 can be read off from the **factorisation**.

**Example 2: Solving a Quadratic Equation by Factorisation (a\neq1)**

Solve by **factorisation** 5x^{2}+54x-11=0

**[3 marks]**

**Step 1:** Rearrange – in this case it is not necessary to do so as the **equation** is already in the form we want.

**Step 2:** **Factorise** the **quadratic**:

5x^{2}+54x-11=(5x-1)(x+11)=0

**Step 3:** Set **both brackets** to 0 and solve:

5x-1=0\rightarrow x=\dfrac{1}{5} and x+11=0\rightarrow x=-11

So the **solutions** are:

x=\dfrac{1}{5} and x=-11

**Example 3: Factorisation and Quadratic Graphs**

Sketch the **graph** y=x^{2}-x-6

**[4 marks]**

**Step 1:** This **quadratic** **graph** is already in the form we are looking for. The **quadratic** we want to solve is x^{2}-x-6=0

**Step 2:** **Factorise** the **quadratic**:

x^{2}-x-6=(x-3)(x+2)=0

**Step 3:** Solve the **equation** by setting **both brackets** to 0:

x-3=0\rightarrow x=3 and x+2=0\rightarrow x=-2

Alternatively, the **solutions** x=3 and x=-2 can be read off from the **factorisation**.

**Step 4:** These **roots** are the x-intercepts of the **graph**, so we know the **graph** passes through the points (3,0) and (-2,0)

**Step 5:** The y-intercept is the number on the end of the **quadratic**, in this case it is -6, so the **graph** passes through (0,-6)

**Step 6:** Sketch the **graph**, making sure it is the right shape and it crosses the axes at the right points.

## Solving Quadratic Equations by Factorisation Example Questions

**Question 1:** Solve, by factorisation, x^{2}-5x+4=0

**[2 marks]**

**Step 1:** Rearrange if necessary – this equation is already in the correct form.

**Step 2:** Factorise the quadratic:

**Step 3:** Set both brackets to 0 and solve:

x-4=0\rightarrow x=4 and x-1=0\rightarrow x=1

Alternatively, we can read the solutions x=4 and x=1 off from the factorisation.

**Question 2:** Solve by factorisation: x^{2}=21x+72

**[3 marks]**

**Step 1:** Rearrange the equation. In this case we need to subtract 21x+72 from both sides:

**Step 2:** Factorise the quadratic:

**Step 3:** Set both brackets to 0 and solve:

x-24=0\rightarrow x=24 and x+3=0\rightarrow x=-3

Alternatively, we can read the solutions x=24 and x=-3 off from the factorisation.

**Question 3:** Solve by factorisation 2x^{2}+11x+12=0

**[3 marks]**

**Step 1:** Rearrange if necessary – this equation is already in the correct form.

**Step 2:** Factorise the quadratic:

**Step 3:** Set both brackets to 0 and solve:

2x+3=0\rightarrow x=-\dfrac{3}{2} and x+4=0\rightarrow x=-4

So the solutions are:

x=-\dfrac{3}{2} and x=-4

**Question 4:** Solve, by factorisation:

**[4 marks]**

**Step 1:** Rearrange the equation. First multiply by 3:

Now subtract 21+2x from both sides:

3x^{2}-2x+21=0**Step 2:** Factorise the quadratic:

**Step 3:** Set both brackets to 0 and solve:

3x+7=0\rightarrow x=-\dfrac{7}{3} and x-3=0\rightarrow x=3

So the solutions are:

x=-\dfrac{7}{3} and x=3

**Question 5:** Sketch the graph y=25x^{2}-25x+6

**[5 marks]**

**Step 1:** This quadratic graph is already in the form we are looking for. The quadratic we want to solve is 25x^{2}-25x+6=0

**Step 2:** Factorise the quadratic:

**Step 3:** Solve the equation by setting both brackets to 0:

5x-2=0\rightarrow x=0.4 and 5x-3=0\rightarrow x=0.6

**Step 4:** These roots are the x-intercepts of the graph, so we know the graph passes through the points (0.4,0) and (0.6,0)

**Step 5:** The y-intercept is the number on the end of the quadratic, in this case it is 6, so the graph passes through (0,6)

**Step 6:** Sketch the graph, making sure it is the right shape and it crosses the axes at the right points.

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